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Theorem of Non-Returning and Time Irreversibility of Tachyon Kinematics.

1 Introduction

Subject of constructing the theory of super-light movement, had been posed in the papers [1,2] more than 50 years ago. Despite the fact that on today tachyons (ie objects moving at a velocity greater than the velocity of light) are not experimentally detected, this subject remains being actual.

It is well known that among physicists it is popular the belief that the hypothesis of the existence of tachyons leads to temporal paradoxes, connected with the possibility of changing the own past. Conditions of appearing these time paradoxes were carefully analyzed in [3]. It should be noted, that in [3] superluminal motion is allowed only for particles or signals whereas superluminal motion for reference frames is forbidden. This fact does not give the possibility to bind the own time with tachyon particle, and, therefore to determine real direction of motion of the particle. In the paper [4] for tachyon particles the own reference frames are axiomatically introduced only for the case of one space dimension. Such approach allows to determine real direction of motion of the tachyon particle by more correct way, and so to obtain more precise results.

In particular, in the paper [4] it was shown, that the hypothesis of existence of material objects, moving with the velocity, greater than the velocity of light, does not lead to formal possibility of returning to the own past in general. Meanwhile in the papers of E. Recami, V. Olkhovsky and R. Goldoni [5-7], and and later in the papers of S. Medvedev [8] as well as J. Hill and B. Cox [9] the generalized Lorentz transforms for superluminal reference frames are deduced in the case of three-dimension space of geometric variables. In the paper [10] it was proven, that the above generalized Lorentz transforms may be easy introduced for the more general case of arbitrary (in particular infinity) dimension of the space of geometric variables.

Further, in [11], using theory of kinematic changeable sets, on the basis of the transformations [10], the mathematically strict models of kinematics, allowing the superluminal motion for particles as well as for inertial reference frames, had been constructed. Thus, the tachyon kinematics in the sense of E.Recami, V. Olkhovsky and R. Goldoni are surely

mathematically strict objects. But, these kinematics are impossible to analyze on the subject of time irreversibility (that is on existence the formal possibility of returning to the own past), using the results of the paper [4], because in [4] complete, multidimensional superluminal reference frames are missing.

Moreover, it can be proved, that the axiom "AxSameFuture" from [4, subsection 2.1] for these tachyon kinematics is not satisfied. The paper [12]1 is based on more general mathematical apparatus in comparison with the paper [4], namely on mathematical apparatus of the theory of kinematic changeable sets. In [12] the strict definitions of time reversibility and time irreversibility for universal kinematics were given, moreover in this paper it was proven, that all tachyon kinematics, constructed in the paper [11], are time reversible in principle. In connection with the last fact the following question arises:

Is it possible to build the certainly time-irreversible universal kinematics, which allows for reference frames moving with any speed other than the speed of light, using the generalized Lorentz-Poincare transformations in terms of E. Recami, V. Olkhovsky and R. Goldoni?

In the present paper we prove the abstract theorem on non-returning for universal kinematics and, using this theorem, we give the positive answer on the last question.

For further understanding of this paper the main concepts and denotation system of the theories of changeable sets, kinematic sets and universal kinematics, are needed. These theories were developed in [11,13-17]. Some of these papers were published in Ukrainian. That is why, for the convenience of readers, main results of these papers were "converted" into English and collected in the preprint [18], where one can find the most complete and detailed explanation of these theories. Hence, we refer to [18] all readers who are not familiar with the essential concepts. So, during citation of needed main results we sometimes will give the dual reference of these results (in one of the papers [11,13-17] as well as in [18]).

1 Note, that main results of the paper [12] were announced in [19].

2 Elementary-time states and changeable systems of universal kinematics

Definition 1. Let F be any universal kinematics (1), l [member of] Lk (F) be any reference frame of F and [omega] [member of] Bs(l) be any elementary-time state in the reference frame l. The set

[[omega].sup.{l,F}] = {(m, <! m [left arrow] l) [omega]) | m [member of] Lk (F)}

(where (x,y) is the ordered pair, composed of x and y) is called by elementary-time state of the universal kinematics F, generated by [omega] in the reference frame I.

Remark 1. In the case, where the universal kinematics F is known in advance, we use the abbreviated denotation [omega]{l} instead of the denotation [[omega].sup.{l,F}].

Assertion 1. Let F be any universal kinematics and l, m [member of] Lk (F). Then for arbitrary elementary-time states [omega] [member of] Bs(l) and [[omega].sub.1] [member of] [B.sub.s](m) the following assertions are equivalent: 1) [[omega].sup.{l}] = [m.sup.{m}.sub.1]; 2) [[omega].sub.1] = <! m [left arrow] l> [omega].

Proof. 1. First, we prove, that statement 2) leads to the statement 1). Consider any [omega] [member of] Bs(l) and [[omega].sub.1] [member of] Bs(m) such that M' = <! m [left arrow] l>[omega]. Applying Definition 1 and [[18, Property 1.12.1(3)].sup.2] , we deduce

[mathematical expression not reproducible]

2. Inversely, suppose, that [omega] [member of] Bs(l), [[omega].sub.1] [member of] Bs(m) and [[omega].sup.{l}] = [[omega].sup.{m}.sub.1]. Then, by Definition 1, we have

{(p, <! p [left arrow] I> [omega]) | p [member of] Lk (F)} = = {(p, <! p [left arrow] m> [[omega].sub.1]) | p [member of] Lk (F)}. (1)

According to [18, Property 1.12.1(1)], we have, <! I [left arrow] I> [omega] = [omega]. Hence, in accordance with (1), for element (l, [omega]) = (l, <! l [left arrow] l>!) [member of] f(p; <! p [left arrow] l>[omega]) | p [member of] Lk (F)} we obtain the correlation, (l, [omega]) [member of] {(p,<! p [left arrow] m>[[omega].sub.1]) | p [member of] Lk (F)}. Therefore, there exists the reference frame [p.sub.0] [member of] Lk (F) such that (l, [omega]) = ([p.sub.0], <! [p.sub.0] [left arrow] m) [[omega].sub.1]). Hence we deduce l = [p.sub.0], as well [omega] = <! [p.sub.0] [left arrow] m) [[omega].sub.1] = <!l [left arrow] m> [[omega].sub.1]. So, based on [18, Properties 1.12.1(1,3)], we conclude, [[omega].sub.1] = <! m [left arrow] I> [[omega].sub.1] = <! m [left arrow] l><!1 [left arrow] m) [[omega].sub.1] = <! m [left arrow] l> [omega].

The next corollary follows from Assertion 1.

Corollary 1. Let F be any universal kinematics. Then for every l, m [member of] Lk (F) and [omega] [member of] Bs(l) the following equality holds:

[omega]{l} = [(<! m [left arrow] l> [omega]).sup.{m}].

(1) Definition of universal kinematics can be found in [11, page 89] or [18, page 156].

(2) Reference to Property 1.12.1(3) means reference to the item 3 from the group of properties "Properties 1.12.1".

Assertion 2. Let F be any universal kinematics. Then the set

Bs [l, F] = {[m.sup.{l,F)] | [omega] [member of] Bs(l)} (2)

does not depend of the reference frame I [member of] Lk (F) (ie VI, m e Lk (F) Bs [I, F] = Bs [m, F]).

Proof. Consider arbitrary I, m e Lk (F). Using Corollary 1, we have

Bs [I, F] = {m{i)| m [member of] Bs(I)} = = {[(<! m [left arrow] I) [omega]).sup.{m}] | [omega] [member of] Bs(l)}.

Hence, according to [18, Corollary 1.12.6], we obtain

Bs [I, F] = {[(<! m [left arrow] I) [omega]).sup.{m}] | [omega] [member of] Bs(I)} = = {[m.sup.{m}.sub.1] | [[omega].sub.1] [member of] Bs(m)} = Bs [m, F].

Definition 2. Let F be any universal kinematics.

1. The set Bs(F) = Bs [I, F] ([for all] I [member of] Lk (F)) is called by the set of all elementary-time states of F.

2. Any subset [??] [subset or equal to] Bs(F) is called by the (common) changeable system of the universal kinematics F.

Assertion 3. Let F be any universal kinematics and I e Lk (F) be any reference frame of F. Then for every element [??] [member of] Bs(F) only one element [[omega].sub.0] [member of] Bs(I) exists such, that [??] = [[??].sup.{l}.sub.0]

Proof. Consider any l [member of] Lk (F) and M [member of] Bs(F). By Definition 2 and Assertion 2 (formula (2)), we have

Bs(F) = Bs [I, F] = {[[omega].sup.{l}]|[omega] [member of] Bs(l)}.

So, since [??] [member of] Bs(F), the element [[omega].sub.0] [member of] Bs(l) must exist such that the following equality is performed:

[??] = [[omega].sup.{l}.sub.0]. (3)

Let us prove that such element [[omega].sub.0] is unique. Assume that [??] = [[omega].sup.{l}.sub.1], where [[omega].sub.1] [member of] Bs(I). Then, from the equality (3) we deduce, [[omega].sup.{l}.sub.0] = [[omega].sup.{l}.sub.1]. Hence, according to Assertion 1 and [18, Property 1.12.1(1)], we obtain, [[omega].sub.1] = <!l [left arrow] l> [[omega].sub.0] = [[omega].sub.0].

Definition 3. Let F be any universal kinematics, [??] [member of] Bs(F) be any elementary-time state of F and l [member of] Lk (F) be any reference frame of F. Elementary-time state [omega] [member of] Bs(I) is named by image of elementary-time state M in the reference frame l if and only if [??] = [omega]{l}.

In accordance with Assertion 3, every elementary-time state [??] [member of] Bs(F) always has only one image in any reference frame l [member of] Lk (F). Image of elementary-time state [??] [member of] Bs(F) in the reference frame l [member of] Lk (Z) will be denoted via [[??].sub.{l,F}] (in the cases, where the universal kinematics F is known in advance, we use the abbreviated denotation [[??].sub.{l}].

Thus, according to Definition 3, for arbitrary [??] [member of] Bs(F) the following equality holds:

[([[??].sub.{l}]).sup.{l}] = [??]. (4)

From the other hand, if for any reference frame l [member of] Lk (F) and any fixed elementary-time state [omega] [member of] Bs(l), we denote [??] := [g.sup.{l}], then by Definition 3, we will receive, [omega] = [[??].sub.{l}].

Therefore we have:

[([[omega].sup.{l}]).sub.{l}] = [omega] ([for all]l [member of] Lk (F) [for all][omega] [member of] Bs[l]). (5)

From equalities (4) and (5) we deduce the following corollary:

Corollary 2. Let F be any universal kinematics and I e Lk (F) be any reference frame of F. Then:

1. The mapping [(*).sup.{l}] is bijectionfrom Bs(I) onto Bs(F).

2. The mapping [(*).sub.{l}] is bijectionfrom Bs(F) onto Bs(I).

3. The mapping [(*).sub.{l}] is inverse to the mapping [(*).sup.{l}].

Assertion 4. Let F be any universal kinematics and {, m e Lk (F) be any reference frames F. Then the following statements are performed:

1. For every [??] [member of] Bs(F) the equality [[??].sub.{m}] = (! m [left arrow] l> [[??].sub.{l}] holds.

2. For each g e Bs({) the equality (g{{}){ } = (1 m ^ {) g is true.

Proof. 1) Chose any [??] [member of] Bs(F). Applying Corollary 1 to the elementary-time state [[??].sub.{l}] [member of] Bs(l) and using equality (4), we obtain

[(<! m [left arrow] l> [[??].sub.{l}]).sup.{m}] = [([[??].sub.{l}]).sup.{l}] = [??]

Thence, using equality (5), we have

[[??].sub.{m}] = [(([(! m [left arrow] I> [[??].sub.{l}]).sup.{m}]).sub.{m}] = <1 m [left arrow] l> [[??].sub.{l}].

2) Consider any [omega] [member of] Bs(l). Applying Corollary 1 as well as equality (5), we deliver

[([[omega].sup.{l}]).sub.{m}] = [([(<! m [left arrow] l>[omega]).sup.{m}]).sub.{m}] = <! m [left arrow] l> [omega].

Let F be any universal kinematics. The set [mathematical expression not reproducible] is called image of changeable system [??] [subset or equal to] Bs(F) in the reference frame l [member of] Lk (F).

Any changeable system A [subset or equal to] Bs(l) in the reference frame l [member of] Lk (F) always generates the (common) changeable system [A.sup.{l,F}] := {[[omega].sup.{l,F}]|[omega] [member of] A} [subset or equal to] Bs(F). Remark 2. In the cases, where universal kinematics F is known in advance, we use the abbreviated denotations [[??].sub.{l}] and [A.sup.{l}] instead of [[??].sub.{l,F}] and [A.sup.{l,F}] (correspondingly).

Applying equalities (4) and (5), we obtain the equalities:

[([[??].sub.{l}]).sup.{l}] = [??] and [([A.sup.{l}]).sub.{l}] = A

(for arbitrary universal kinematics F, reference frame l [member of] Lk (F) and changeable systems [??] [subset or equal to] Bs(F) as well A [subset or equal to] Bs(l)).

3 Chain paths of universal kinematics and definition of time irreversibility

Definition 4. Let F be any universal kinematics. Changeable system [??] [subset or equal to] Bs(F) is called piecewise chain changeable system if and only if there exist the sequences of changeable systems [[??].sub.1], ..., [[bar.A].sub.n] [subset or equal to] Bs(F) and reference frames [l.sub.1], ..., [l.sub.n] [member of] Lk (F) (n [member of] N) satisfying the following conditions:

(a) [mathematical expression not reproducible] where definition of set Ll ([l.sub.k]) = Ll (([l.sub.k])[conjunction]) can be found in [18, pages 63, 88, 156];

(b) [[union].sup.n.sub.k=1] [[??].sub.k] = [??],

and, moreover, in the case n [greater than or equal to] 2 the following additional conditions are satisfied:

(c) [[??].sub.k] [intersection [[??].sub.k+1] [not equal to] 0 ([for all]k [member of] [bar.1, n - 1]);

(d) For each k [member of] 1, [n.sup.-1] and arbitrary [mathematical expression not reproducible] the inequality [mathematical expression not reproducible] holds.

(e) For every k [member of] [bar.2, n] and arbitrary [mathematical expression not reproducible] the inequality [mathematical expression not reproducible] is performed.

In this case Jhe ordered composition [mathematical expression not reproducible], will be named by the chain path of universal kinematics F.

Definition 5. Let F be any universal kinematics.

(a) Changeable system A [subset or equal to] Bs(l) is refereed to as geometrically-stationary in the reference frame l [member of] Lk (F) if and only if A [member of] Ll(l) and for arbitrary [[omega].sub.1], [[omega].sub.2] [member of] A the equality bs ([Q.sup.<l>])([[omega].sub.1])) = bs ([Q.sup.<l>] = [[omega].sub.2])) holds.

(b) The set of all geometrically-stationary changeable sys tems in the reference frame {is denoted via Lg(l, F). In the cases, where the universal kinematics F is known in advance, we use the abbreviated denotation Lg(l).

(c) The chain path [mathematical expression not reproducible] in F (n [member of] N) is called by piecewise geometrically-stationary if and only if [mathematical expression not reproducible].

From the physical point of view piecewise geometricallystationary chain path may be interpreted as process of "vagrancy" of observer (or some material particle or signal), which moves by means of "jumping" from previous reference frame to the next frame with a finite number of times.

Definition 6. Let F be any universal kinematics and let [mathematical expression not reproducible], be arbitrary chain path in F.

1. Element [[??].sub.s] [member of] Bs(F) is called by start element of the path ., if and only if [mathematical expression not reproducible] and for every [mathematical expression not reproducible] the inequality [mathematical expression not reproducible] is performed.

2. Element [[??].sub.f] [member of] Bs(F) is called by final element of the path A, if and only if [[??].sub.f] [member of] [[??].sub.n] and for every [mathematical expression not reproducible] the inequality [mathematical expression not reproducible] holds.

3. The chain path A, which owns (at least one) start element and (at least one) final element, is called by closed.

Assertion 5. Any chain path A of arbitrary universal kinematics F can not have more, than one start element and more, than one final element.

Proof. (a) Let [[??].sub.s], [[??].sub.x] be two start elements of the chain path [mathematical expression not reproducible]. Then, by Definition 6, we have [mathematical expression not reproducible]. Therefore we get

[mathematical expression not reproducible]. (6)

Since [mathematical expression not reproducible], where, in accordance with Definition 4 (subitem (a)), we have, [mathematical expression not reproducible]. That is, according to [18, Assertion 1.7.5 (item 1)], [mathematical expression not reproducible] is a function from Tm (I') into Bs (I'). So, using equality m = (tm ([omega]), bs ([omega])) ([omega] [member of] Bs ([I.sub.l])) as well as formula (6), we obtain

[mathematical expression not reproducible].

Using the last equality and equality (6), we deduce, [mathematical expression not reproducible]. Hence, according to formula (4), we deliver [mathematical expression not reproducible].

(c) Similarly it can be proven that the chain path A can not have more, than one final element.

Further the start element of the chain path A of the universal kinematics F will be denoted via po (A, F), or via po (A). The final element of the chain path A will be denoted via ki (A, F), or via ki (A). Where the denotations po (A) and ki (A) are used in the cases when they do not cause misunderstanding. Thus, for every closed chain path A both start and final elements (po (A) and ki (A)) always exist.

Definition 7. Closed chain path A of universal kinematics F is refereed to as geometrically-cyclic in the reference frame l [member of] Lk (F) if and only if bs ([Q.sup.<l>] (po [(A).sub.{l}])) = bs ([Q.sup.<l>] (ki [(A).sub.{l}])).

Definition 8. Universal kinematics F is called time irreversible if and only if for every reference frame I [member of] Lk (F) and for each chain path A, geometrically-cyclic in the frame I andpiecewise geometrically-stationary in F, it is performed the inequality tm (po (A){l}) [[less than or equal to].sub.l] tm (ki [(A).sub.{l}]).

Universal kinematics F is called time reversible if and only if it is not time irreversible.

The physical sense of time irreversibility notion is that in time irreversible kinematics there is not any process or object which returns to the begin of the own path at the past, moving by means of "jumping" from previous reference frame to the next frame. So, there are not temporal paradoxes in these kinematics.

4 Direction of time between reference frames of universal kinematics

For formulation main theorem we need some notions, connected with direction of time between reference frames.

Definition 9. Let F be any universal kinematics.

1. We say that reference frame m [member of] Lk (F) is timenonnegative relatively the reference frame I [member of] Lk (F) (in the universal kinematics F) (denotation is m [[??].sub.F] l) if and only if for arbitrary [w.sub.1], [w.sub.2] [member of] Mk (l) such that bs([w.sub.1]) = bs ([w.sub.2]) and tm ([w.sub.1]) [[less than or equal to].sub.l] tm ([w.sub.2]) it is performed the inequality, tm ([m [left arrow] I][w.sub.1]) [[less than or equal to].sub.m] tm ([m [left arrow] l] [w.sub.2]).

2. We say that reference frame m [member of] Lk (F) is timepositive in F relatively the reference frame I [member of] Lk (F) (denotation is m [[??].sup.+.sub.F] l) if and only if for arbitrary [w.sub.1], [w.sub.2] [member of] Mk (I) such that bs ([w.sub.1]) = bs ([w.sub.2]) and tm ([w.sub.1]) [<.sub.l] tm ([w.sub.2]) it is performed the inequality, tm ([m ^ I] [w.sub.1]) [<.sub.m] tm ([m [left arrow] I] [w.sub.2]).

3. We say that reference frame m [member of] Lk (F) is time-nonpositive in F relatively the reference frame l [member of] Lk (F) (denotation is m [[??].sub.F] I) if and only if for arbitrary w', [w.sub.2] [member of] Mk (I) such that bs (w) = bs ([w.sub.2]) and tm (w) [[less than or equal to].sub.l] tm ([w.sub.2]) it is performed the inequality, tm ([m [left arrow] l] [w.sub.1]) [[greater than or equal to].sub.m] tm ([m [left arrow] I] [w.sub.2]).

4. We say that reference frame m [member of] Lk (F) is timenegative in F relatively the reference frame l [member of] Lk (F) (denotation is m [[??].sup.-.sub.F] l) if and only if for arbitrary [w.sub.1], [w.sub.2] [member of] Mk (l) such that bs (w) = bs ([w.sub.2]) and tm ([w.sub.1]) [<.sub.l] tm ([w.sub.2]) it is performed the inequality, tm ([m [left arrow] I] [w.sub.1]) [>.sub.m] tm ([m [left arrow] I] [w.sub.2]).

5. The universal kinematics F is named by weakly timepositive if and only if there exist at least one reference frame [l.sub.0] [member of] Lk (F) such that the correlation [l.sub.0] [[??].sup.+.sub.F] l holds for every reference frame l [member of] Lk (F).

Remark 3. Apart from weak time-positivity we can introduce other, more strong, form of time-positivity. We say that universal kinematics F is time-positive if and only if for arbitrary reference frames I, m e Lk (F) the correlation I m holds. It is not hard to prove that every kinematics of kind F = UP (H, B, c) (connected with classical special relativity and introduced in [11] and [18, Section 24]) is time-positive.

Assertion 6. For arbitrary reference frames l, m [member of] Lk (F) of any universal kinematics F the following statements are performed.

1) If m [[??].sup.+.sub.F] I, then m [[??].sub.F] I.

2) If m [[??].sup.-.sub.F], then m [[??].sup.+.sub.F].

Proof. 1) Indeed, let m [member of] Lk (F) and m [[??].sup.+.sub.F]. Then for every [w.sub.1]; [w.sub.2] [member of] Mk (l) such, that bs ([w.sub.1]) = bs ([w.sub.2]) and tm ([w.sub.1]) [[less than or equal to].sub.l] tm ([w.sub.2]), we deduce the following:

(a) In the case tm ([w.sub.1]) <{ tm ([w.sub.2]), by Definition 9, item 2, we get, tm ([m [left arrow] l] [w.sub.1]) [<.sub.m] tm ([m [left arrow] l] [w.sub.2]).

(b) In the case tm ([w.sub.1]) = tm ([w.sub.2]), we have [w.sub.1] = (tm ([w.sub.1]), bs ([w.sub.1])) = (tm ([w.sub.2]), bs ([w.sub.2])) = [w.sub.2], and so tm ([m [left arrow] I] [w.sub.1]) = tm ([m [left arrow] I] [w.sub.2]).

2) Second item of this Assertion can be proven similarly.

5 Theorem of Non-Returning

Theorem 1. Any weakly time-positive universal kinematics F is time irreversible.

To prove Theorem 1 we need a few auxiliary assertions.

Assertion 7. Let [??] [subset or equal to] Bs(F) be changeable system of universal kinematics F such, that [mathematical expression not reproducible] for some reference frame [I.sub.0] [member of] Lk (F). Let I [member of] Lk (F) be reference frame, satisfying condition I [[??].sub.F] [I.sub.0].

Then for arbitrary [mathematical expression not reproducible] the inequality [mathematical expression not reproducible] assures the the inequality [mathematical expression not reproducible].

Proof. Suppose that, unde(r condi)tions of(the ass)ertion, we have [mathematical expression not reproducible]. According to Definition of Minkowski coordinates (see [11, formula (2)] or [18, formula (2.3)]), we have [mathematical expression not reproducible]. So, we get

[mathematical expression not reproducible]. (7)

Since [mathematical expression not reproducible] then, by Definition 5 (items (a),(b)), we have

[mathematical expression not reproducible]. (8)

Taking into account that {!0 and using Definition 9 (item 1) as well as formulas (7), (8), we get the inequality:

[mathematical expression not reproducible].

Thence, using [18, formula (3.2)], we obtain

[mathematical expression not reproducible].

Applying the last inequality as well as Assertion 4, we deduce the inequality:

[mathematical expression not reproducible]. (9)

According to Definition of Minkowski coordinates (see [11, formula (2)] or [18, formula (2.3)]), for every [omega] [member of] Bs(I) we have the equality tm ([Q.sup.<I>]([omega])) = tm ([omega]). That is why from the inequality (9) it follows the desired inequality [mathematical expression not reproducible].

Assertion 8. Let, [mathematical expression not reproducible] be closed, piecewise geometrically-stationary chain path of universal kinematics F and I [member of] Lk (F) be reference frame such that I [[??].sub.F] [I.sub.i] for every i [member of] [bar.1,n]. Then for arbitrary [mathematical expression not reproducible] the following inequality holds:

[mathematical expression not reproducible]. (10)

Proof Let F be universal kinematics and A = [mathematical expression not reproducible] be closed, piecewise geometrically-stationary chain path of F. Let, I [member of] Lk (F) be reference frame such that I [[??].sub.F] [I.sub.i] ([for all] i [member of] [bar.1,n]).

1) First we prove that for any [mathematical expression not reproducible] it holds the inequality:

[mathematical expression not reproducible]. (11)

By Definition 4 (item (b)), [??] = [[union].sup.n.sub.k=1] [[??].sub.k]. So, it is sufficient to prove the inequality (11) for the cases [mathematical expression not reproducible].

1.a) First we prove the inequality (11) for [mathematical expression not reproducible]. According to Definition 6 (item 1), for [mathematical expression not reproducible] we obtain that po (A) [member of] [[??].sup.1] and

[mathematical expression not reproducible]. (12)

According to the above, we have [mathematical expression not reproducible]. Moreover, by Definition 5 (item (c)), we get, [mathematical expression not reproducible]. By conditions of Assertion, we have, I [[??].sub.F] [I.sub.1]. So, in accordance with Assertion 7, the correlation (12) stipulates the inequality tm (po ([A.sub).{I}]) [[less than or equal to].sub.I] tm ([[??].sub.{I}]). Hence, in the case [mathematical expression not reproducible], the inequality (11) has been proven. Moreover, the last inequality has been proven for all [mathematical expression not reproducible] in the case n = 1. So, further we consider, that n > 1.

1.b) Assume, that inequality (11) is performed for all [??] [member of] [[??].sub.k-1], where k [member of] [bar.2,n]. And, let us prove, that then this inequality is true for each [mathematical expression not reproducible].

In the case [mathematical expression not reproducible] the inequality (11) is true in accordance with inductive hypothesis. Hence, it remains to prove the last inequality for every [mathematical expression not reproducible]. According to item (c) of Definition 4, we have [[??].sub.k] [intersection] [[??].sub.k-1] [not equal to] 0. Hence, at least one element [mathematical expression not reproducible] exists. Since,

[mathematical expression not reproducible], (13)

then we get [mathematical expression not reproducible]. Therefore, according to item (e) of Definition 4, we deliver

[mathematical expression not reproducible]. (14)

According to (13), we have [mathematical expression not reproducible], where, by item (c) of Definition 5, [mathematical expression not reproducible]. Since I [[??].sub.F] [I.sub.k], then taking into account inequality (14) and Assertion 7 we deduce

[mathematical expression not reproducible]. (15)

According to (13), we have [mathematical expression not reproducible]. So, by inductive hypothesis, we deliver

[mathematical expression not reproducible]. (16)

Inequalities (15) and (16) assure inequality (11).

Thus, by Principle of mathematical induction, inequality (11) is true for arbitrary [mathematical expression not reproducible].

2) Now we are aiming to prove, that for any MM e A it holds the inequality:

[mathematical expression not reproducible]. (17)

2.a) First we prove the inequality (17) for [omega] [member of] [[??].sub.n]. According to Definition 6 (item 2), for [mathematical expression not reproducible] we obtain that ki (A) [member of] [[??].sub.n] and

[mathematical expression not reproducible]. (18)

According to the above, we have [mathematical expression not reproducible]. Moreover, by Definition 5 (item (c)), we get [mathematical expression not reproducible]. By conditions of Assertion, we have I [??]F [I.sub.n]. So, in accordance with Assertion 7, the correlation (18) stipulates the inequality (17). Hence, in the case [mathematical expression not reproducible], the inequality (17) is proven. Moreover, the last inequality is proven for all [mathematical expression not reproducible] in the case n = 1. So, further we consider, that n > 1.

2.b) Assume, that inequality (17) is performed for all [mathematical expression not reproducible], where k [member of] 1, [bar.n - 1]. And, let us prove, that then this inequality is true for each [mathematical expression not reproducible].

In the case [omega] [member of] [[??].sub.k] [intersection] [[??].sub.k+1] the inequality (17) is true in accordance with inductive hypothesis. Hence, it remains to prove the last inequality for every [mathematical expression not reproducible]. According to item (c) of Definition, we have [mathematical expression not reproducible]. Hence, at least one element [mathematical expression not reproducible]1 exists. Taking into account that

[mathematical expression not reproducible], (19)

we get [mathematical expression not reproducible]. Therefore, according to item (d) of Definition 4, we deliver

[mathematical expression not reproducible]. (20)

According to (19), we have [mathematical expression not reproducible] by item (c) of Definition 5. Since I [[??].sub.F] [I.sub.k] then, taking into account inequality (20) and Assertion 7, we deduce

[mathematical expression not reproducible]. (21)

According to (19), we have [mathematical expression not reproducible]. So, by inductive hypothesis, we deliver

[mathematical expression not reproducible]. (22)

Inequalities (21) and (22) assure inequality (17). Thus, by Principle of mathematicalinduction, inequality (17) is true for arbitrary [mathematical expression not reproducible].

Inequality (10) follows from (11) and (17).

Proof of Theorem 1. Let F be weakly time-positive universal kinematics. Then, by Definition 9, there exists the reference frame [I.sub.0] [member of] Lk (F) such that

[for all]m [member of] Lk(F) [I.sub.0] [[??].sup.+.sub.F] m. (23)

Let [mathematical expression not reproducible] be piecewise geometrically-stationary chain path in F and, moreover, A is geometrically-cyclic relatively some reference frame I [member of] Lk (F). By Definition 7, A is closed chain path. According to Assertion 6, correlation (23) leads to the correlation [I.sub.0] [[??].sub.F] [I.sub.k] ([for all]k [member of] [bar.1,n]). Hence, applying Assertion 8, we ensure

[mathematical expression not reproducible]. (24)

Assume, that tm (ki [(A).sub.{I}]) [<.sub.I] tm (po [(A).sub.{I}]. Then, by Definition of Minkowski coordinates (see [11, formula (2)] or [18, formula (2.3)]), we obtain

[mathematical expression not reproducible]. (25)

Since the path A is geometrically-cyclic relatively the reference frame l, then, by Definition 7, we have

[mathematical expression not reproducible]. (26)

Since (in accordance with (23)) [I.sub.0] [[??].sup.+.sub.F] l, then, by Definition 9 (item 2), from the correlations (25), and (26), we get the inequality:

[mathematical expression not reproducible].

Thence, using [18, formula (3.2)], we deduce the inequality:

[mathematical expression not reproducible].

Taking into account Assertion 4, the last inequality can be reduced to the form, [mathematical expression not reproducible], and, by Definition of Minkowski coordinates (see [11, formula (2)] or [18, formula (2.3)])), we assure

[mathematical expression not reproducible].

But, the last inequality contradicts to the (correla)tion (24). Therefore, hypothesis affirming, that tm (ki [(A).sub.{I}]}) [<.sub.I] tm (po [(A).sub.{I}]) is false. Consequently we have

tm (po [(A).sub.{I}]) [[less than or equal to].sub.I] tm (ki [(A).sub.{I}]). (27)

Thus, for each reference frame I [member of] Lk (F) and for each chain path A, geometrically-cyclic in the frame I and piecewise geometrically-stationary in F, it holds the inequality (27). So, by Definition 8, kinematics F is time irreversible, which must be proved.

6 Certainly time irreversibility. Strengthened version of theorem of non-returning

Recall, that in the papers [17, Definition 6], [18, Definition 3.25.2] the notion of equivalence of universal kinematics relatively coordinate transform had been introduced. According to these papers, we denote equivalent relatively coordinate transform kinematics [F.sub.1] and [F.sub.2] via [F.sub.1] [[equivalent to]] F2.

Definition 10. We say that universal kinematics F is certainly time irreversible if and only if arbitrary universal kinematics [F.sub.1] such, that F [[equivalent to]] [F.sub.1] is time irreversible. In the opposite case we will say that universal kinematics F is conditionally time reversible.

Since, according to [17, Assertion 3] (see also [18, Assertion 3.25.1]), for each universal kinematics F it is fulfilled the correlation F [[equivalent to]] F, then we receive the following Corollary from Definition 10:

Corollary 3. Any certainly time irreversible universal kinematics F is time irreversible.

The physical sense of certain time irreversibility notion is that in certainly time irreversible kinematics temporal paradoxes are impossible basically, that is there is not potential possibility to affect the own past by means of "traveling" and "jumping" between reference frames. Whereas, in time irreversible, but conditionally time reversible kinematics such potential possibility exists, but it is not realized in the scenario of evolution, acting in this kinematics.

Assertion 9. Let universal kinematics F be weakly time-positive. Then every universal kinematics F1 such that [F.sub.1] [[equivalent to]] F is weakly time-positive also.

Proof. Let F be weakly time-positive universal kinematics and [F.sub.1] [[equivalent to]] F. Recall, that in [18, Definition 3.27.3] for every reference frame m [member of] Lk (F) it was introduced the reference frame [mathematical expression not reproducible], related with m in the universal kinematics [F.sub.1]:

[mathematical expression not reproducible]. (28)

Since kinematics F is weakly time-positive then, by Definition 9, the reference frame [I.sub.0] [member of] Lk (F) exists such that for each reference frame I [member of] Lk (F) the correlation [I.sub.0] [[??].sup.+.sub.F] {holds. Denote:

[mathematical expression not reproducible].

Let us consider any reference frame [I.sup.(1)] [member of] Lk ([F.sub.1]). Denote: I := [I.sup.(1)] [|.sub.F] [member of] Lk(F). Then, according to [18, Properties 3.27.1] and formula (28), we have

[mathematical expression not reproducible].

Hence, taking into account [18, Definition 3.25.2 (item 2)], formula (28) and [18, Property 3.25.1(1)], we get

[mathematical expression not reproducible]. (29)

Similarly applying [18, Definition 3.25.2 (item 2)] we ensure the equalities:

Tm ([I.sup.(1).sub.0]) = Tm ([I.sub.0]); Tm ([I.sup.(1)]) = Tm (I) (30)

(where (in accordance with [18, Subsection 6.3]) Tm(m) = (Tm (m), [[less than or equal to].sub.m]) ([for all] m [member of] Lk (F) [union] Lk ([F.sub.1]))). Moreover, according to [18, Property 3.25.1(1) and Definition 3.25.2 (item 3)], we obtain

[mathematical expression not reproducible]. (31)

Taking into account (29(), let) us consider any elements [w.sub.1], [w.sub.2] [member of] Mk ([I.sup.(1)]; [F.sub.1]) = Mk ({I; F) such that bs ([w.sub.1]) = bs ([w.sub.2]) and [mathematical expression not reproducible]. Then, in accordance with (30), we obtain the inequality tm ([w.sub.1]) [<.sub.I] tm ([w.sub.2]). Since (as it was mentioned before) {0 {, then, by Definition 9 (item 2), we obtain the inequality tm [mathematical expression not reproducible]. Thence, using (31) and (30), we ensure the inequality, [mathematical expression not reproducible]. By Definition 9 (item 2), taking into account the arbitrariness of choice elements [w.sub.1], [w.sub.2] [member of] Mk ([I.sup.(1)]; [F.sub.1]) such, that [mathematical expression not reproducible], we obtain the correlation [mathematical expression not reproducible] (for every reference frame l(1) [member of] Lk ([F.sub.1])). Hence, by Definition 9, kinematics [F.sub.1] is weakly time-positive.

Applying Assertion 9 as well as Theorem 1, we obtain the following (strengthened) variant of Theorem of Non-Returning:

Theorem 2. Any weakly time-positive universal kinematics F is certainly time irreversible.

7 Example of certainly time irreversible tachyon

kinematics

In this section we build the certainly time-irreversible universal kinematics, which allows for reference frames moving with any speed other than the speed of light, using the generalized Lorentz-Poincare transformations in terms of E. Recami, V. Olkhovsky and R. Goldoni.

Let (H, [parallel]h[parallel], <*, *>) be a Hilbert space over the real field such, that dim(h) [greater than or equal to] 1, where dim(H) is dimension of the space H. Emphasize, that the condition dim(H) [greater than or equal to] 1 should be interpreted in a way that the space h may be infinite-dimensional. Let L (H) be the space of (homogeneous) linear continuous operators over the space h. Denote by Lx (H) the space of all operators of affine transformations over the space H, that is [L.sup.x] (h) = {[A.sub.[a]] | A [member of] L (H), a [member of] H}, where [A.sub.[a]]x = Ax + a, x e H. The Minkowski space over the Hilbert space H is defined as the Hilbert space M (H) = R x h = {(t, x) | t [member of] R, x [member of] h}, equipped by the inner product and norm: [mathematical expression not reproducible]. In the space M (h) we select the next subspaces: [h.sub.0] := {(t, 0) | t [member of] R}, [h.sup.1] := {(0, x) | x [member of] h} with 0 being zero vector. Then, M (h) = [h.sub.0] [direct sum] [h.sub.1], where [direct sum] means the orthogonal sum of subspaces. Denote: [e.sub.0] := (1,0) e M(h). Introduce the orthogonal projectors on the subspaces H1 and [h.sub.0]:

[mathematical expression not reproducible].

Let B1 (H1) be the unit sphere in the space [h.sub.1] ([B.sub.1] ([h.sub.1]) = {x [member of] [h.sub.1] | [parallel]x[parallel] = 1}). Any vector n [member of] [B.sub.1] ([h.sub.1]) generates the following orthogonal projectors, acting in M (H):

[X.sub.1] [n] w = <n, w> n (w [member of] M (H)); [X.sup.[perpendicular to].sub.1] [n] = X - [X.sup.1] [n].

Recall, that an operator U [member of] L (H) is referred to as unitary on H, if and only if [mathematical expression not reproducible]. Let U ([h.sub.1]) be the set of all unitary operators over the space [h.sub.1].

Fix some real number c such, that 0 < c < [infinity]. Denote:

[mathematical expression not reproducible], (32)

where[mathematical expression not reproducible] are operators of generalized Lorentz-Poincare Transformations in the sense of E. Recami, V. Olkhovsky and R. Goldoni, introduced in [10,11,18]:

[mathematical expression not reproducible]. (33)

According to [18,20], every operator of kind [W.sub.[lambda],c] [s, n, J; a] belongs to Pk (H), where Pk (H) is the set of all operators S [member of] [L.sup.x] (M (H)), which have the continuous inverse operator [S.sup.-1] [member of] [L.sup.x] (M (H)). Using results of the papers [18,20], we can calculate the operators, inverse to the operators of kind [W.sub.[lambda],c] [s, n, J] and [W.sub.[lambda],c] [s, n, J; a].

Lemma 1. For arbitrary c [member of] (0, [infinity]), [lambda] [member of] [0, [infinity]) \ {c}, s [member of] {-1,1}, J [member of] U ([h.sub.1]) and n [member of] [B.sub.1] ([h.sub.1]) the following equality holds:

[mathematical expression not reproducible]. (34)

Proof. Consider arbitrary 0 < c < [infinity], [lambda] [member of] [0, [infinity]) \ {c}, s [member of] {-1,1}, J [member of] U ([h.sub.1]) and n [member of] [B.sub.1] ([h.sub.1]). According to [10, page 143] or [18, formula (2.86)], operator [W.sub.[lambda],c] [s, n, J] may be represented in the form:

[W.sub.[lambda],c] [s, n, J] = [U.sub.[theta],c] [s, n, J], (35)

where

[mathematical expression not reproducible].

Hence, according to [20, Corollary 5.1] or [18, Corollary 2.18.3], we obtain, that [([W.sub.[lambda],c] [s, n, J]).sup.-1] [member of] L (M (H)), and moreover:

[mathematical expression not reproducible]. (36)

In the case s = 1 we have, [s.sub.[theta]] = sign [theta] = sign [mathematical expression not reproducible] sign (c-[lambda]). Hence, in this case, using (36) and (35), we obtain

[mathematical expression not reproducible]. (37)

Now we consider the case s = -1 ([[theta].sup.s] = [[theta].sup.-1]). Applying (36) and [18, formula (2.90)], in this case we deduce

[mathematical expression not reproducible]. (38)

Taking into account (37) and (38) in the both cases we obtain (34).

Using Lemma 1, we obtain the following corollary.

Corollary 4. For arbitrary c [member of] (0, [infinity]), [lambda] [member of] [0, [infinity])\{c}, s [member of] {-1,1}, J [member of] U ([h.sub.1]), n [member of] [B.sub.1] ([h.sub.1]) and a [member of] M (H) the following equality is fulfilled:

[mathematical expression not reproducible].

Let B be any base changeable set such, that Bs(B) [subset or equal to] H and Tm(B) = (R, [less than or equal to]), where [less than or equal to] is the standard order in the field of real numbers R. Denote:

[mathematical expression not reproducible], (39)

where the denotation Ku (*,*; *) is introduced in [11], [18, page 166]. From [18, Assertion 2.17.5] it follows, that in the case dim(h) = 3 universal kinematics [UBI.sup.[??].sub.fin] (h, B, c) may be considered as tachyon extension of kinematics of classical special relativity, which allows for reference frames moving with arbitrary speed other than the speed of light.

According to [18, Property 3.23.1(1)], the set Lk ([UBI.sup.[??].sub.fin] (H, B, c)) of all reference frames of universal kinematics [UBI.sup.[??].sub.fin] (H, B, c), defined by (39), can be represented in the form:

[mathematical expression not reproducible]. (40)

In accordance with [18, Corollary 2.19.5], subclass of operators

[mathematical expression not reproducible]

is group of operators over the space M (H). So, the identity operator [I.sub.M(H)]w = w ([for all] w [member of] M (H)) belongs to the class [BI.sup.[??].sub.fin] (H, c). Hence, in accordance with (40), we may define the following reference frame:

[mathematical expression not reproducible] (41)

(recall, that, according to [18, Remark 1.11.3], [I.sub.M(h)] [B] = B).

Lemma 2. For each reference frame I [member of] Lk([UBI.sup.[??].sub.fin] (h, B, c) the following correlation holds:

[mathematical expression not reproducible].

Proof. Consider any reference frame I [member of] Lk([UBI.sup.[??].sub.fin] (h, B, c)). According to (40) and (32), frame I can be represented in the form:

I = (U, U [B]), where (42)

[mathematical expression not reproducible]. (43)

Applying [18, Properties 3.23.1(3,4,7)] as well (42), (43), (41) and Corollary 4 we obtain

[mathematical expression not reproducible]; (44)

[mathematical expression not reproducible]. (45)

Now we consider any [w.sub.1], [w.sub.2] [member of] Mk (I) = M (H) such that bs ([w.sub.1]) = bs ([w.sub.2]) and tm ([w.sub.1]) [<.sub.I] tm ([w.sub.2]). According to (44), inequality tm ([w.sub.1]) [<.sub.I] tm ([w.sub.2]) is equivalent to the inequality tm ([w.sub.1]) < tm ([w.sub.2]). From the equality bs ([w.sub.1]) = bs ([w.sub.2]) it

follows that

[mathematical expression not reproducible].

Thence, using (45) and (33) we deduce

[mathematical expression not reproducible]

Therefore, tm ([[I.sub.0,B] [left arrow] I][w.sub.1] < tm ([[I.sub.0,B] [left arrow] I] [w.sub.2]), ie, according to (44), we have, [mathematical expression not reproducible]. Thus, for arbitrary [w.sub.1], [w.sub.2] [member of] Mk (I) = M (H) such, that bs ([w.sub.1]) = bs ([w.sub.2]) and tm ([w.sub.1]) < tm ([w.sub.2]) it is true the inequality [mathematical expression not reproducible]. And, taking into account Definition 9 (item 2), we have seen, that [mathematical expression not reproducible].

Corollary 5. Every universal kinematics of kind [UBI.sup.[??].sub.fin] (H, B, c) (0 < c < [infinity]) is certainly time irreversible.

Proof. According to Lemma 2 and Definition 9 (item 5), kinematics of kind [UBI.sup.[??].sub.fin] (h, B, c) (0 < c < [infinity]) is weakly time-positive. Hence, by Theorem 2, kinematics [UBI.sup.[??].sub.fin] (h, B, c) is certainly time irreversible.

Remark 4. Kinematics of kind [UBI.sup.[??].sub.fin] (h, B, c) (0 < c < [infinity]) is weakly time-positive, but it is not time-positive. Similarly to Lemma 2 it can be proved, that for any (superluminal) reference frame of kind:

[mathematical expression not reproducible]

the correlation [mathematical expression not reproducible] is true despite the fact that [mathematical expression not reproducible] (according to Lemma 2).

Remark 5. It is easy to see that the binary relation [[??].sup.+.sub.F] is reflexive on the set Lk (F) of all reference frames of arbitrary universal kinematics F. From Remark 4 it follows that in the general case this relation is not symmetric. Using the results of [10, Section 7, paragraph 4] it can be proven that this relation is not transitive in the general case.

8 On the physical interpretation of main result

The aim of this section is to explain main Theorem 2 in the physical language. We can imagine, that any universal kinematics F is some abstract "world", which not necessarily coincides with the our. In every such "world" F there exists the fixed for this "world" set of reference frames Lk (F). We reach the agreement that for any reference frame I [member of] Lk (F) the arrows of the clock, fixed in the frame I are rotating clockwise relatively the frame I. We say, that the reference frame m [member of] Lk (F) is time-positive relatively the reference frame I [member of] Lk (F) (ie m [[??].sup.+.sub.F] I) if and only if the observer in the reference frame m (fixed relatively m) observes that the arrows of the clock, fixed in the frame I are rotating clockwise in the frame m as well (cf. Definition 9, item 2). We abandon the physical question, how can the observer in m "see" the clock, fixed in the other frame I. From the mathematical point of view, the possibility of observation the clock, attached to another reference frame, is guaranteed by existence of universal coordinate transform between every two reference frames (see definition of universal kinematics in [11,18]). According to Remark 5, the binary relation [[??].sup.+.sub.F] always is reflexive, but, in the general case, it is not symmetric and is not transitive on the set Lk (F) of all reference frames of the "world" F.

We also suppose, that in the "world" F the interframe voyagers can exist. Such voyagers may move from one reference frame to the another frame, passing near them (similarly as, standing near the tram track, we can jump into the tram, passing near us).

From the physical point of view Theorem 2 asserts, that if in the "world" F there exists at least one reference frame [I.sub.0] [member of] Lk (F), which is time-positive relatively the every frame I [member of] Lk (F), then in this "world" the temporal paradoxes, connected with the possibility of the returning to the own past are impossible. This means, that any interframe voyager, starting in some reference frame I in some fixed point x can not finish its travel in the frame I and in the point x at the past time.

9 Conclusions

1. According to Corollary 5, kinematics of kind [UBI.sup.[??].sub.fin] (H, B, c) (in the case dim(H) = 3) gives the example of certainly time-irreversible tachyon extension of kinematics of classical special relativity, which allows for reference frames moving with arbitrary velocity other than the velocity of light. Thus, the main conclusion of Theorem 2 is the following:

In the general case the hypothesis of existence of material objects and inertial reference frames, moving with the velocity, greater than the velocity of light, does not lead to temporal paradoxes, connected with existence of formal possibility of returning to the own past.

2. In [9] authors have deduced two variants of generalized superluminal Lorentz transforms for the case, when two inertial frames are moving along the common xaxis:

[mathematical expression not reproducible], (46)

where v [member of] R, [absolute value of (v)] > c (see [9, formula (3.16)]) and:

[mathematical expression not reproducible]7, (47)

(see [9, formula (3.18)]). Transforms (46) are particular cases of the transforms of kind (33) for the case, where dim(H) = 3, [lambda] > c and s = 1, whereas transforms (47) belong to the transforms of kind (33) for the case, where dim(H) = 3, [lambda] > c and s = -1. If we chose in (33) the value s = 1 for subluminal as well as superluminal diapason, we obtain the class of operators [BI.sub.+] (h, c), defined in [13,18] and based on this class of operators universal kinematics of kind UPT (H, B, c). According to results, announced in [19] and published in [12], this kinematics is conditionally time reversible (1a). But, if we chose in (33) the value s = 1 for subluminal diapason and value s = -1 for superluminal diapason, we reach the class of operators UBI (H, c), defined in (32) and based on this class of operators universal kinematics of kind UPfi (H, B, c). According to Corollary 5, kinematics UpTfn (H, B, c) is certainly time irreversible. Thus we can formulate the following conclusion, concerning two variants of superluminal Lorentz transforms, deduced in [9]: From the standpoint of time-irreversibility, transforms (47) or [9, formula (3.18)] are more suitable for representation of the tachyon continuation of Einstein's special theory of relativity than (46) or [9, formula (3.16)].

Main results of this paper had been announced in [19].

Received on September 19, 2017

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[2.] Bilaniuk O.-M. P., Sudarshan E.C.G. Particles beyond the light barrier. Physics Today, 1969, v. 22, no. 5, 43-51.

[3.] V.I. Ilyashevytch, S.Y. Medvedev. Tachyon acausal loops. Uzhhorod University Scientific Herald. Series Physics, 2010, no. 27, 103-111 (in Ukrainian).

[4.] Andreka H., Madarasz J.X., Nemeti I., Stannett M., Szekely G. Faster than light motion does not imply time travel. Classical and Quantum Gravity, 2014, v. 31, no. 9, 095005.

[5.] Recami E., Olkhovsky V.S. About Lorentz transformations and tachyons. Lettere al Nuovo Cimento, 1971, v. 1, no. 4, 165-168.

[6.] Goldoni R. Faster-than-light inertial frames, interacting tachyons and tadpoles. Lettere al Nuovo Cimento, 1972, v. 5, no. 6, 495-502.

[7.] Recami E. Classical Tachyons and Possible Applications. Riv. Nuovo Cimento, 1986, v. 9, s.3, no. 6, 1-178.

[8.] Medvedev S.Yu. On the possibility of broadening special relativity theory beyond light barrier. Uzhhorod University Scientific Herald. Ser. Phys., 2005, no. 18, 7-15 (in Ukrainian).

[9.] Hill J.M., Cox B.J. Einstein's special relativity beyond the speed of light. Proc. of the Royal Society A, 2012, v. 468, no. 2148, 4174-4192.

[10.] Grushka Ya.I. Tachyon generalization for lorentz transforms. Methods of Functional Analysis and Topology, 2013, v. 20, no. 2, 127-145.

[11.] Grushka Ya.I. Kinematic changeable sets with given universal coordinate transform. Proceedings of Institute of Mathematics NAS of Ukraine (Zb. Pr. Inst. Mat. NAN Ukr.), 2015, v. 12, no. 1, 74-118 (in Ukrainian).

[12.] Grushka Ya.I. On time reversibility of tachyon kinematics. Proceedings of Institute of Mathematics NAS of Ukraine (Zb. Pr. Inst. Mat. NAN Ukr.], 2016, v. 13, no. 2, 125-174 (in Ukrainian).

[13.] Grushka Ya.I. Changeable sets and their application for the construction of tachyon kinematics. Proceedings of Institute of Mathematics NAS of Ukraine (Zb. Pr. Inst. Mat. NAN Ukr.), 2014, v. 11, no. 1, 192-227 (in Ukrainian).

[14.] Grushka Ya.I. Existence criteria for universal coordinate transforms in kinematic changeable sets. Bukovinian Mathematical Journal, 2014, v. 2, no. 2-3, 59-71 (in Ukrainian).

[15.] Grushka Ya.I. Abstract concept of changeable set. Preprint: arXiv:1207.3751v1, (2012), 54 p.

[16.] Grushka Ya.I. Coordinate transforms in kinematic changeable sets. Reports of the National Academy of Sciences of Ukraine, 2015, no. 3, 2431, (in Ukrainian).

[17.] Grushka Ya.I. Evolutionary extensions of kinematic sets and universal kinematics. Proceedings of Institute of Mathematics NAS of Ukraine (Zb. Pr. Inst. Mat. NAN Ukr.), 2015, v. 12, no. 2, 139-204 (in Ukrainian).

[18.] Grushka Ya.I. Draft introduction to abstract kinematics. (Version 2.0). Preprint: viXra: 1701.0523v2, (2017), 208 p.

[19.] Grushka Ya.I. On time irreversibility of universal kinematics. Reports of the National Academy of Sciences of Ukraine, 2016, no. 7, 14-21, (in Ukrainian).

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Yaroslav I. Grushka

Institute of Mathematics NAS of Ukraine, 3, Tereschenkivska st., 01601, Kyiv, Ukraine.

(1) Further we denote via [bar.m, n] (m, n [member of] N, m [less than or equal to] n) the set [bar.m, n] = {m, ..., n}.

(1a) In fact, class of operators [BI.sub.+] (H, c) contains apart from operators of kind (33) (with s = 1) also operators, corresponding tachyon inertial reference frames with infinite velocities. However, using results of the paper [12], it is not hard to deduce that the "subkinematics" of kinematics UBI (h, B, c), which includes only all reference frames from UBI (H, B, c) with finite velocities, also is conditionally time reversible.
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